Nematic liquid crystals are typically described either by a unit vector field $n$ (Oseen-Frank model, OF) or a symmetric, traceless order parameter tensor field $Q$ (Landau-de Gennes model, LdG). It is a well-known fact that the standard LdG description reduces to the general OF approach in the limit of vanishing nematic correlation length only if elastic terms up to the cubic order in $Q$ are taken into account. The LdG energy that results has elastic contribution that is not bounded from below over the set of admissible tensor fields. Because at the same time, the nematic-to-isotopic transition cannot be described within the OF theory, various modifications to the LdG theory have been suggested that include, for example, more general potentials motivated by the mean-field theories.

I will describe a model based on collecting appropriate fourth-order elastic terms from the generalized LdG theory proposed by Longa and collaborators in the 1980s. The corresponding variational model uses the standard LdG potential, is well-posed, and reduces to the general OF model when the nematic correlation length tends to zero. The model allows us to consider nematic-to-isotopic transitions for highly disparate elastic constants. Numerical simulations demonstrate that theoretical predictions match well with experimental observations.

This is joint work with Michael Novack and Peter Sternberg (Indiana), Oleg Lavrentovich (Kent State), and Young-Ki Kim (Cornell)